Brain fog and atypical depression are common symptoms of CFS/IBS/FM. There was a recent study that readers sent me that is a good kick-start for this post.
Differences in gut microbial composition correlate with regional brain volumes in irritable bowel syndrome . [Full Text]” Gut microbial composition correlated with structural measures of brain regions including sensory- and salience-related regions, and with a history of early life trauma.”
- “Examining correlations in female IBS only revealed that Bacteroidales positively correlated with overall symptom severity”
- “For example, the Firmicutes-associated Clostridia (higher in IBS1) and the Bacteroidetes-associated Bacteroidia (lower in IBS1) showed correlations with the volume of several subcortical brain regions involved in sensory integration and modulation and the motor cortex. For the majority of these regions, increased volumes were observed with decreases in Bacteroidia taxa and increases in the Clostridia taxa characterizing IBS1.”
- And thus light sensitivity, sound sensitivity and possibly multiple chemical sensitivity appear associated with shifts of the microbiome …
- “Probiotic Bifidobacterium longum NCC3001 Reduces Depression Scores and Alters Brain Activity: a Pilot Study in Patients With Irritable Bowel Syndrome .
- “reduces depression but not anxiety scores and increases quality of life in patients with IBS. These improvements were associated with changes in brain activation patterns that indicate that this probiotic reduces limbic reactivity.”
Microbiota modulation counteracts Alzheimer’s disease progression influencing neuronal proteolysis and gut hormones plasma levels .
- “our results clearly prove that modulation of the microbiota induces positive effects on neuronal pathways that are able to slow down the progression of Alzheimer’s disease.”
Structural changes of gut microbiota in Parkinson’s disease and its correlation with clinical features .
- ” The results suggest that structural changes of gut microbiota in PD are characterized by the decreases of putative cellulose degraders and the increases of putative pathobionts, which may potentially reduce the production of short chain fatty acids, and produce more endotoxins and neurotoxins; and these changes is potentially associated with the development of PD pathology.”
The influence of a probiotic supplementation on memory in quail suggests a role of gut microbiota on cognitive abilities in birds.
- “In conclusion, the supplementation with Pediococcus acidilactici as a probiotic, affected a specific trait of emotional reactivity in STI quail, and improved memory in both lines, whichstrengthens the idea that the influence of gut microbiota on the host behaviour and memory seen in mammals is shared by birds.”
- “Furthermore, the role of microbiota in neurodevelopment and in the modulation of social behavior suggests the possibility of its role in autism spectrum disorder and in schizophrenia. If altered, microbiota play a role in psychiatric disorders, then efforts to normalize the gut microbial population by the ingestion of probiotics (live bacteria) could have antidepresssant or antipsychotic effects. Testing such hypotheses in translational human studies is a matter of future research.” 
- “Emerging evidence links perturbations in the gut microbiota to neurological disease, including disease risk, activity, and progression. This review provides an overview of the recent advances in microbiome research in relation to neuro(auto)immune and neurodegenerative conditions affecting humans, such as multiple sclerosis, neuromyelitis optica spectrum disorders, Parkinson disease, Alzheimer disease, Huntington disease, and amyotrophic lateral sclerosis.” 
- Gut to Brain Dysbiosis: Mechanisms Linking Western Diet Consumption, the Microbiome, and Cognitive Impairment .
- ” In some cases treatment with specific probiotics or prebiotics can prevent or reverse some of the deleterious impact of Western Diet consumption on neuropsychological outcomes, indicating that targeting the microbiome may be a successful strategy for combating dietary- and metabolic-associated cognitive impairment.” 
- ZiBuPiYin recipe improves cognitive decline by regulating gut microbiota in Zucker diabetic fatty rats .
- Effects of gut microbiota on the microRNA and mRNA expression in the hippocampus of mice.
- Human gut microbiota: the links with dementia development .
There is growing evidence and a lot of research (note that every citation above is from 2017) that the gut microbiome alters cognitive function and behaviors. In terms of documented probiotics influencing cognitive function, Bifidobacterium longum NCC3001, would be a good start. I suspect most of the Bifidobacterium longum would be beneficial.
In my own experience, I have noticed that significant cognitive changes/improvement when I rotated to some probiotics. It does not happen always — and sometimes it only happens on the initial taking of the probiotic.